A fundamental result on equation (REF ) is the celebrated Liouville-type theorem due to Gidas and Spruck [1]}. The equation (REF ) has no positive classical solution if \(1<p<\frac{N+2}{N-2}.\) 
Moreover, it was also proved that the exponent \(\frac{N+2}{N-2}\)  is optimal, in the sense that problem (REF ) admits a positive solution for \(p\ge \frac{N+2}{N-2}\)  and \(N\ge 3\) . Soon afterward, similar results were established in [2]} for positive solutions of the subcritical problem (REF ) in the upper half-space \(\mathbb {R}^N_+\) .
These results received wide attention as regards the theory itself and its applications. Particularly, when variational methods cannot be employed, one use them to establish a prior bound of solutions for general operator, and therefore existence of solutions may be dealt with via topological methods; see for instance [1]}, [2]}.
